X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=5f058ca9a8c055f972ffffaa216975517e0dc723;hb=9bab3c89cc7d669e7b99295900c9f590e5525079;hp=baff1a7bbea4943206bc323040ef5d554d40ead4;hpb=cbd0e9a0ed9133de08665a55f146d04cb9536ec6;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index baff1a7..5f058ca 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -8,117 +8,41 @@ addsitedir(abspath('../../')) from sage.all import * -def _basically_the_same(K1, K2): +def _restrict_to_space(K, W): r""" - Test whether or not ``K1`` and ``K2`` are "basically the same." - - This is a hack to get around the fact that it's difficult to tell - when two cones are linearly isomorphic. We have a proposition that - equates two cones, but represented over `\mathbb{Q}`, they are - merely linearly isomorphic (not equal). So rather than test for - equality, we test a list of properties that should be preserved - under an invertible linear transformation. - - OUTPUT: - - ``True`` if ``K1`` and ``K2`` are basically the same, and ``False`` - otherwise. - - EXAMPLES: - - Any proper cone with three generators in `\mathbb{R}^{3}` is - basically the same as the nonnegative orthant:: - - sage: K1 = Cone([(1,0,0), (0,1,0), (0,0,1)]) - sage: K2 = Cone([(1,2,3), (3, 18, 4), (66, 51, 0)]) - sage: _basically_the_same(K1, K2) - True - - Negating a cone gives you another cone that is basically the same:: - - sage: K = Cone([(0,2,-5), (-6, 2, 4), (0, 51, 0)]) - sage: _basically_the_same(K, -K) - True - - TESTS: - - Any cone is basically the same as itself:: - - sage: K = random_cone(max_dim = 8) - sage: _basically_the_same(K, K) - True - - After applying an invertible matrix to the rows of a cone, the - result should be basically the same as the cone we started with:: - - sage: K1 = random_cone(max_dim = 8) - sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') - sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) - sage: _basically_the_same(K1, K2) - True - - """ - if K1.lattice_dim() != K2.lattice_dim(): - return False - - if K1.nrays() != K2.nrays(): - return False - - if K1.dim() != K2.dim(): - return False - - if K1.lineality() != K2.lineality(): - return False - - if K1.is_solid() != K2.is_solid(): - return False - - if K1.is_strictly_convex() != K2.is_strictly_convex(): - return False - - if len(LL(K1)) != len(LL(K2)): - return False - - C_of_K1 = discrete_complementarity_set(K1) - C_of_K2 = discrete_complementarity_set(K2) - if len(C_of_K1) != len(C_of_K2): - return False - - if len(K1.facets()) != len(K2.facets()): - return False - - return True - - - -def _rho(K, K2=None): - r""" - Restrict ``K`` into its own span, or the span of another cone. + Restrict this cone a subspace of its ambient space. INPUT: - - ``K2`` -- another cone whose lattice has the same rank as this - cone. + - ``W`` -- The subspace into which this cone will be restricted. OUTPUT: - A new cone in a sublattice. + A new cone in a sublattice corresponding to ``W``. + + EXAMPLES: - EXAMPLES:: + When this cone is solid, restricting it into its own span should do + nothing:: sage: K = Cone([(1,)]) - sage: _rho(K) == K + sage: _restrict_to_space(K, K.span()) == K True + A single ray restricted into its own span gives the same output + regardless of the ambient space:: + sage: K2 = Cone([(1,0)]) - sage: _rho(K2).rays() + sage: K2_S = _restrict_to_space(K2, K2.span()).rays() + sage: K2_S N(1) in 1-d lattice N sage: K3 = Cone([(1,0,0)]) - sage: _rho(K3).rays() + sage: K3_S = _restrict_to_space(K3, K3.span()).rays() + sage: K3_S N(1) in 1-d lattice N - sage: _rho(K2) == _rho(K3) + sage: K2_S == K3_S True TESTS: @@ -126,417 +50,101 @@ def _rho(K, K2=None): The projected cone should always be solid:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K_S = _rho(K) - sage: K_S.is_solid() + sage: K = random_cone(max_ambient_dim = 8) + sage: _restrict_to_space(K, K.span()).is_solid() True And the resulting cone should live in a space having the same dimension as the space we restricted it to:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K_S = _rho(K, K.dual() ) - sage: K_S.lattice_dim() == K.dual().dim() + sage: K = random_cone(max_ambient_dim = 8) + sage: K_P = _restrict_to_space(K, K.dual().span()) + sage: K_P.lattice_dim() == K.dual().dim() True This function should not affect the dimension of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.dim() == _rho(K).dim() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.dim() == _restrict_to_space(K,K.span()).dim() True Nor should it affect the lineality of a cone:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.lineality() == _rho(K).lineality() + sage: K = random_cone(max_ambient_dim = 8) + sage: K.lineality() == _restrict_to_space(K, K.span()).lineality() True No matter which space we restrict to, the lineality should not increase:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8) - sage: K.lineality() >= _rho(K).lineality() + sage: K = random_cone(max_ambient_dim = 8) + sage: S = K.span(); P = K.dual().span() + sage: K.lineality() >= _restrict_to_space(K,S).lineality() True - sage: K.lineality() >= _rho(K, K.dual()).lineality() + sage: K.lineality() >= _restrict_to_space(K,P).lineality() True If we do this according to our paper, then the result is proper:: sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() + sage: K = random_cone(max_ambient_dim = 8) + sage: K_S = _restrict_to_space(K, K.span()) + sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual() sage: K_SP.is_proper() True - sage: K_SP = _rho(K_S, K_S.dual()) + sage: K_SP = _restrict_to_space(K_S, K_S.dual().span()) sage: K_SP.is_proper() True - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() - sage: K_SP.is_proper() - True - sage: K_SP = _rho(K_S, K_S.dual()) - sage: K_SP.is_proper() - True - - Test Proposition 7 in our paper concerning the duals and + Test the proposition in our paper concerning the duals and restrictions. Generate a random cone, then create a subcone of - it. The operation of dual-taking should then commute with rho:: - - sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=False) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) - True - - :: - - sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=False) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) - True - - :: - - sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=False, strictly_convex=True) - sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) - sage: _basically_the_same(K_W_star, K_star_W) - True - - :: + it. The operation of dual-taking should then commute with + _restrict_to_space:: sage: set_random_seed() - sage: J = random_cone(max_dim = 8, solid=True, strictly_convex=True) + sage: J = random_cone(max_ambient_dim = 8) sage: K = Cone(random_sublist(J.rays(), 0.5), lattice=J.lattice()) - sage: K_W_star = _rho(K, J).dual() - sage: K_star_W = _rho(K.dual(), J) + sage: K_W_star = _restrict_to_space(K, J.span()).dual() + sage: K_star_W = _restrict_to_space(K.dual(), J.span()) sage: _basically_the_same(K_W_star, K_star_W) True """ - if K2 is None: - K2 = K - - # First we project K onto the span of K2. This will explode if the - # rank of ``K2.lattice()`` doesn't match ours. - span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice()) - K = K.intersection(span_K2) - - # Cheat a little to get the subspace span(K2). The paper uses the - # rays of K2 as a basis, but everything is invariant under linear - # isomorphism (i.e. a change of basis), and this is a little - # faster. - W = span_K2.linear_subspace() + # First we want to intersect ``K`` with ``W``. The easiest way to + # do this is via cone intersection, so we turn the subspace ``W`` + # into a cone. + W_cone = Cone(W.basis() + [-b for b in W.basis()], lattice=K.lattice()) + K = K.intersection(W_cone) # We've already intersected K with the span of K2, so every # generator of K should belong to W now. - W_rays = [ W.coordinate_vector(r) for r in K.rays() ] - - L = ToricLattice(K2.dim()) - return Cone(W_rays, lattice=L) - - - -def discrete_complementarity_set(K): - r""" - Compute the discrete complementarity set of this cone. - - The complementarity set of a cone is the set of all orthogonal pairs - `(x,s)` such that `x` is in the cone, and `s` is in its dual. The - discrete complementarity set is a subset of the complementarity set - where `x` and `s` are required to be generators of their respective - cones. - - For polyhedral cones, the discrete complementarity set is always - finite. - - OUTPUT: - - A list of pairs `(x,s)` such that, - - * Both `x` and `s` are vectors (not rays). - * `x` is a generator of this cone. - * `s` is a generator of this cone's dual. - * `x` and `s` are orthogonal. - - REFERENCES: - - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. - - EXAMPLES: - - The discrete complementarity set of the nonnegative orthant consists - of pairs of standard basis vectors:: - - sage: K = Cone([(1,0),(0,1)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((0, 1), (1, 0))] - - If the cone consists of a single ray, the second components of the - discrete complementarity set should generate the orthogonal - complement of that ray:: - - sage: K = Cone([(1,0)]) - sage: discrete_complementarity_set(K) - [((1, 0), (0, 1)), ((1, 0), (0, -1))] - sage: K = Cone([(1,0,0)]) - sage: discrete_complementarity_set(K) - [((1, 0, 0), (0, 1, 0)), - ((1, 0, 0), (0, -1, 0)), - ((1, 0, 0), (0, 0, 1)), - ((1, 0, 0), (0, 0, -1))] - - When the cone is the entire space, its dual is the trivial cone, so - the discrete complementarity set is empty:: - - sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) - sage: discrete_complementarity_set(K) - [] - - Likewise when this cone is trivial (its dual is the entire space):: - - sage: L = ToricLattice(0) - sage: K = Cone([], ToricLattice(0)) - sage: discrete_complementarity_set(K) - [] - - TESTS: - - The complementarity set of the dual can be obtained by switching the - components of the complementarity set of the original cone:: - - sage: set_random_seed() - sage: K1 = random_cone(max_dim=6) - sage: K2 = K1.dual() - sage: expected = [(x,s) for (s,x) in discrete_complementarity_set(K2)] - sage: actual = discrete_complementarity_set(K1) - sage: sorted(actual) == sorted(expected) - True - - The pairs in the discrete complementarity set are in fact - complementary:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=6) - sage: dcs = discrete_complementarity_set(K) - sage: sum([x.inner_product(s).abs() for (x,s) in dcs]) - 0 - - """ - V = K.lattice().vector_space() - - # Convert rays to vectors so that we can compute inner products. - xs = [V(x) for x in K.rays()] - - # We also convert the generators of the dual cone so that we - # return pairs of vectors and not (vector, ray) pairs. - ss = [V(s) for s in K.dual().rays()] - - return [(x,s) for x in xs for s in ss if x.inner_product(s) == 0] - - -def LL(K): - r""" - Compute the space `\mathbf{LL}` of all Lyapunov-like transformations - on this cone. - - OUTPUT: - - A list of matrices forming a basis for the space of all - Lyapunov-like transformations on the given cone. - - EXAMPLES: - - The trivial cone has no Lyapunov-like transformations:: - - sage: L = ToricLattice(0) - sage: K = Cone([], lattice=L) - sage: LL(K) - [] - - The Lyapunov-like transformations on the nonnegative orthant are - simply diagonal matrices:: - - sage: K = Cone([(1,)]) - sage: LL(K) - [[1]] - - sage: K = Cone([(1,0),(0,1)]) - sage: LL(K) - [ - [1 0] [0 0] - [0 0], [0 1] - ] - - sage: K = Cone([(1,0,0),(0,1,0),(0,0,1)]) - sage: LL(K) - [ - [1 0 0] [0 0 0] [0 0 0] - [0 0 0] [0 1 0] [0 0 0] - [0 0 0], [0 0 0], [0 0 1] - ] - - Only the identity matrix is Lyapunov-like on the `L^{3}_{1}` and - `L^{3}_{\infty}` cones [Rudolf et al.]_:: - - sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) - sage: LL(L31) - [ - [1 0 0] - [0 1 0] - [0 0 1] - ] - - sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) - sage: LL(L3infty) - [ - [1 0 0] - [0 1 0] - [0 0 1] - ] - - If our cone is the entire space, then every transformation on it is - Lyapunov-like:: - - sage: K = Cone([(1,0), (-1,0), (0,1), (0,-1)]) - sage: M = MatrixSpace(QQ,2) - sage: M.basis() == LL(K) - True - - TESTS: - - The inner product `\left< L\left(x\right), s \right>` is zero for - every pair `\left( x,s \right)` in the discrete complementarity set - of the cone:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: C_of_K = discrete_complementarity_set(K) - sage: l = [ (L*x).inner_product(s) for (x,s) in C_of_K for L in LL(K) ] - sage: sum(map(abs, l)) - 0 - - The Lyapunov-like transformations on a cone and its dual are related - by transposition, but we're not guaranteed to compute transposed - elements of `LL\left( K \right)` as our basis for `LL\left( K^{*} - \right)` - - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: LL2 = [ L.transpose() for L in LL(K.dual()) ] - sage: V = VectorSpace( K.lattice().base_field(), K.lattice_dim()^2) - sage: LL1_vecs = [ V(m.list()) for m in LL(K) ] - sage: LL2_vecs = [ V(m.list()) for m in LL2 ] - sage: V.span(LL1_vecs) == V.span(LL2_vecs) - True - - """ - V = K.lattice().vector_space() - - C_of_K = discrete_complementarity_set(K) - - tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] - - # Sage doesn't think matrices are vectors, so we have to convert - # our matrices to vectors explicitly before we can figure out how - # many are linearly-indepenedent. - # - # The space W has the same base ring as V, but dimension - # dim(V)^2. So it has the same dimension as the space of linear - # transformations on V. In other words, it's just the right size - # to create an isomorphism between it and our matrices. - W = VectorSpace(V.base_ring(), V.dimension()**2) - - # Turn our matrices into long vectors... - vectors = [ W(m.list()) for m in tensor_products ] - - # Vector space representation of Lyapunov-like matrices - # (i.e. vec(L) where L is Luapunov-like). - LL_vector = W.span(vectors).complement() - - # Now construct an ambient MatrixSpace in which to stick our - # transformations. - M = MatrixSpace(V.base_ring(), V.dimension()) - - matrix_basis = [ M(v.list()) for v in LL_vector.basis() ] - - return matrix_basis + K_W_rays = [ W.coordinate_vector(r) for r in K.rays() ] + L = ToricLattice(W.dimension()) + return Cone(K_W_rays, lattice=L) def lyapunov_rank(K): r""" - Compute the Lyapunov (or bilinearity) rank of this cone. + Compute the Lyapunov rank of this cone. - The Lyapunov rank of a cone can be thought of in (mainly) two ways: - - 1. The dimension of the Lie algebra of the automorphism group of the - cone. - - 2. The dimension of the linear space of all Lyapunov-like - transformations on the cone. - - INPUT: - - A closed, convex polyhedral cone. + The Lyapunov rank of a cone is the dimension of the space of its + Lyapunov-like transformations -- that is, the length of a + :meth:`lyapunov_like_basis`. Equivalently, the Lyapunov rank is the + dimension of the Lie algebra of the automorphism group of the cone. OUTPUT: - An integer representing the Lyapunov rank of the cone. If the - dimension of the ambient vector space is `n`, then the Lyapunov rank - will be between `1` and `n` inclusive; however a rank of `n-1` is - not possible (see the first reference). - - .. note:: + A nonnegative integer representing the Lyapunov rank of this cone. - In the references, the cones are always assumed to be proper. We - do not impose this restriction. - - .. seealso:: - - :meth:`is_proper` + If the ambient space is trivial, the Lyapunov rank will be zero. + Otherwise, if the dimension of the ambient vector space is `n`, then + the resulting Lyapunov rank will be between `1` and `n` inclusive. A + Lyapunov rank of `n-1` is not possible [Orlitzky]_. ALGORITHM: @@ -549,21 +157,21 @@ def lyapunov_rank(K): REFERENCES: - .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of a proper - cone and Lyapunov-like transformations, Mathematical Programming, 147 - (2014) 155-170. + .. [Gowda/Tao] M.S. Gowda and J. Tao. On the bilinearity rank of + a proper cone and Lyapunov-like transformations. Mathematical + Programming, 147 (2014) 155-170. - .. [Orlitzky/Gowda] M. Orlitzky and M. S. Gowda. The Lyapunov Rank of an - Improper Cone. Work in-progress. + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html - .. [Rudolf et al.] G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear - optimality constraints for the cone of positive polynomials, - Mathematical Programming, Series B, 129 (2011) 5-31. + G. Rudolf, N. Noyan, D. Papp, and F. Alizadeh, Bilinear + optimality constraints for the cone of positive polynomials, + Mathematical Programming, Series B, 129 (2011) 5-31. EXAMPLES: The nonnegative orthant in `\mathbb{R}^{n}` always has rank `n` - [Rudolf et al.]_:: + [Rudolf]_:: sage: positives = Cone([(1,)]) sage: lyapunov_rank(positives) @@ -576,7 +184,7 @@ def lyapunov_rank(K): 3 The full space `\mathbb{R}^{n}` has Lyapunov rank `n^{2}` - [Orlitzky/Gowda]_:: + [Orlitzky]_:: sage: R5 = VectorSpace(QQ, 5) sage: gs = R5.basis() + [ -r for r in R5.basis() ] @@ -585,20 +193,20 @@ def lyapunov_rank(K): 25 The `L^{3}_{1}` cone is known to have a Lyapunov rank of one - [Rudolf et al.]_:: + [Rudolf]_:: sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) sage: lyapunov_rank(L31) 1 - Likewise for the `L^{3}_{\infty}` cone [Rudolf et al.]_:: + Likewise for the `L^{3}_{\infty}` cone [Rudolf]_:: sage: L3infty = Cone([(0,1,1), (1,0,1), (0,-1,1), (-1,0,1)]) sage: lyapunov_rank(L3infty) 1 A single ray in `n` dimensions should have Lyapunov rank `n^{2} - n - + 1` [Orlitzky/Gowda]_:: + + 1` [Orlitzky]_:: sage: K = Cone([(1,0,0,0,0)]) sage: lyapunov_rank(K) @@ -607,7 +215,7 @@ def lyapunov_rank(K): 21 A subspace (of dimension `m`) in `n` dimensions should have a - Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky/Gowda]_:: + Lyapunov rank of `n^{2} - m\left(n - m)` [Orlitzky]_:: sage: e1 = (1,0,0,0,0) sage: neg_e1 = (-1,0,0,0,0) @@ -621,7 +229,7 @@ def lyapunov_rank(K): 19 The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + [Rudolf]_:: sage: L31 = Cone([(1,0,1), (0,-1,1), (-1,0,1), (0,1,1)]) sage: octant = Cone([(1,0,0), (0,1,0), (0,0,1)]) @@ -629,7 +237,7 @@ def lyapunov_rank(K): sage: lyapunov_rank(K) == lyapunov_rank(L31) + lyapunov_rank(octant) True - Two isomorphic cones should have the same Lyapunov rank [Rudolf et al.]_. + Two isomorphic cones should have the same Lyapunov rank [Rudolf]_. The cone ``K`` in the following example is isomorphic to the nonnegative octant in `\mathbb{R}^{3}`:: @@ -638,7 +246,7 @@ def lyapunov_rank(K): 3 The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: + itself [Rudolf]_:: sage: K = Cone([(2,2,4), (-1,9,0), (2,0,6)]) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) @@ -647,89 +255,33 @@ def lyapunov_rank(K): TESTS: The Lyapunov rank should be additive on a product of proper cones - [Rudolf et al.]_:: + [Rudolf]_:: sage: set_random_seed() - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: K2 = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K1 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) + sage: K2 = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: K = K1.cartesian_product(K2) sage: lyapunov_rank(K) == lyapunov_rank(K1) + lyapunov_rank(K2) True The Lyapunov rank is invariant under a linear isomorphism - [Orlitzky/Gowda]_:: - - sage: K1 = random_cone(max_dim = 8) - sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') - sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) - sage: lyapunov_rank(K1) == lyapunov_rank(K2) - True - - Just to be sure, test a few more:: - - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') - sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) - sage: lyapunov_rank(K1) == lyapunov_rank(K2) - True + [Orlitzky]_:: - :: - - sage: K1 = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') - sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) - sage: lyapunov_rank(K1) == lyapunov_rank(K2) - True - - :: - - sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=True) - sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') - sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) - sage: lyapunov_rank(K1) == lyapunov_rank(K2) - True - - :: - - sage: K1 = random_cone(max_dim=8, strictly_convex=False, solid=False) + sage: K1 = random_cone(max_ambient_dim = 8) sage: A = random_matrix(QQ, K1.lattice_dim(), algorithm='unimodular') sage: K2 = Cone( [ A*r for r in K1.rays() ], lattice=K1.lattice()) sage: lyapunov_rank(K1) == lyapunov_rank(K2) True The dual cone `K^{*}` of ``K`` should have the same Lyapunov rank as ``K`` - itself [Rudolf et al.]_:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - Make sure we exercise the non-strictly-convex/non-solid case:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - Let's check the other permutations as well, just to be sure:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - :: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) - True - - :: + itself [Rudolf]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8) sage: lyapunov_rank(K) == lyapunov_rank(K.dual()) True @@ -740,7 +292,9 @@ def lyapunov_rank(K): the Lyapunov rank of the trivial cone will be zero:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: (n == 0 or 1 <= b) and b <= n @@ -748,70 +302,51 @@ def lyapunov_rank(K): sage: b == n-1 False - In fact [Orlitzky/Gowda]_, no closed convex polyhedral cone can have + In fact [Orlitzky]_, no closed convex polyhedral cone can have Lyapunov rank `n-1` in `n` dimensions:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: b = lyapunov_rank(K) sage: n = K.lattice_dim() sage: b == n-1 False The calculation of the Lyapunov rank of an improper cone can be - reduced to that of a proper cone [Orlitzky/Gowda]_:: + reduced to that of a proper cone [Orlitzky]_:: sage: set_random_seed() - sage: K = random_cone(max_dim=8) + sage: K = random_cone(max_ambient_dim=8) sage: actual = lyapunov_rank(K) - sage: K_S = _rho(K) - sage: K_SP = _rho(K_S.dual()).dual() + sage: K_S = _restrict_to_space(K, K.span()) + sage: K_SP = _restrict_to_space(K_S.dual(), K_S.dual().span()).dual() sage: l = K.lineality() sage: c = K.codim() sage: expected = lyapunov_rank(K_SP) + K.dim()*(l + c) + c**2 sage: actual == expected True - The Lyapunov rank of a proper cone is just the dimension of ``LL(K)``:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) - True - - In fact the same can be said of any cone. These additional tests - just increase our confidence that the reduction scheme works:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=False) - sage: lyapunov_rank(K) == len(LL(K)) - True - - :: + The Lyapunov rank of a cone is the size of a :meth:`lyapunov_like_basis`:: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=True) - sage: lyapunov_rank(K) == len(LL(K)) + sage: K = random_cone(max_ambient_dim=8) + sage: lyapunov_rank(K) == len(K.lyapunov_like_basis()) True - :: + We can make an imperfect cone perfect by adding a slack variable + (a Theorem in [Orlitzky]_):: sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=False, solid=False) - sage: lyapunov_rank(K) == len(LL(K)) - True - - Test Theorem 3 in [Orlitzky/Gowda]_:: - - sage: set_random_seed() - sage: K = random_cone(max_dim=8, strictly_convex=True, solid=True) + sage: K = random_cone(max_ambient_dim=8, + ....: strictly_convex=True, + ....: solid=True) sage: L = ToricLattice(K.lattice_dim() + 1) sage: K = Cone([ r.list() + [0] for r in K.rays() ], lattice=L) sage: lyapunov_rank(K) >= K.lattice_dim() True """ - beta = 0 + beta = 0 # running tally of the Lyapunov rank m = K.dim() n = K.lattice_dim() @@ -819,19 +354,333 @@ def lyapunov_rank(K): if m < n: # K is not solid, restrict to its span. - K = _rho(K) + K = _restrict_to_space(K, K.span()) - # Lemma 2 - beta += m*(n - m) + (n - m)**2 + # Non-solid reduction lemma. + beta += (n - m)*n if l > 0: # K is not pointed, restrict to the span of its dual. Uses a # proposition from our paper, i.e. this is equivalent to K = # _rho(K.dual()).dual(). - K = _rho(K, K.dual()) + K = _restrict_to_space(K, K.dual().span()) - # Lemma 3 - beta += m * l + # Non-pointed reduction lemma. + beta += l * m - beta += len(LL(K)) + beta += len(K.lyapunov_like_basis()) return beta + + + +def is_lyapunov_like(L,K): + r""" + Determine whether or not ``L`` is Lyapunov-like on ``K``. + + We say that ``L`` is Lyapunov-like on ``K`` if `\left\langle + L\left\lparenx\right\rparen,s\right\rangle = 0` for all pairs + `\left\langle x,s \right\rangle` in the complementarity set of + ``K``. It is known [Orlitzky]_ that this property need only be + checked for generators of ``K`` and its dual. + + INPUT: + + - ``L`` -- A linear transformation or matrix. + + - ``K`` -- A polyhedral closed convex cone. + + OUTPUT: + + ``True`` if it can be proven that ``L`` is Lyapunov-like on ``K``, + and ``False`` otherwise. + + .. WARNING:: + + If this function returns ``True``, then ``L`` is Lyapunov-like + on ``K``. However, if ``False`` is returned, that could mean one + of two things. The first is that ``L`` is definitely not + Lyapunov-like on ``K``. The second is more of an "I don't know" + answer, returned (for example) if we cannot prove that an inner + product is zero. + + REFERENCES: + + M. Orlitzky. The Lyapunov rank of an improper cone. + http://www.optimization-online.org/DB_HTML/2015/10/5135.html + + EXAMPLES: + + The identity is always Lyapunov-like in a nontrivial space:: + + sage: set_random_seed() + sage: K = random_cone(min_ambient_dim = 1, max_rays = 8) + sage: L = identity_matrix(K.lattice_dim()) + sage: is_lyapunov_like(L,K) + True + + As is the "zero" transformation:: + + sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: R = K.lattice().vector_space().base_ring() + sage: L = zero_matrix(R, K.lattice_dim()) + sage: is_lyapunov_like(L,K) + True + + Everything in ``K.lyapunov_like_basis()`` should be Lyapunov-like + on ``K``:: + + sage: K = random_cone(min_ambient_dim = 1, max_rays = 5) + sage: all([ is_lyapunov_like(L,K) for L in K.lyapunov_like_basis() ]) + True + + """ + return all([(L*x).inner_product(s) == 0 + for (x,s) in K.discrete_complementarity_set()]) + + +def random_element(K): + r""" + Return a random element of ``K`` from its ambient vector space. + + ALGORITHM: + + The cone ``K`` is specified in terms of its generators, so that + ``K`` is equal to the convex conic combination of those generators. + To choose a random element of ``K``, we assign random nonnegative + coefficients to each generator of ``K`` and construct a new vector + from the scaled rays. + + A vector, rather than a ray, is returned so that the element may + have non-integer coordinates. Thus the element may have an + arbitrarily small norm. + + EXAMPLES: + + A random element of the trivial cone is zero:: + + sage: set_random_seed() + sage: K = Cone([], ToricLattice(0)) + sage: random_element(K) + () + sage: K = Cone([(0,)]) + sage: random_element(K) + (0) + sage: K = Cone([(0,0)]) + sage: random_element(K) + (0, 0) + sage: K = Cone([(0,0,0)]) + sage: random_element(K) + (0, 0, 0) + + TESTS: + + Any cone should contain an element of itself:: + + sage: set_random_seed() + sage: K = random_cone(max_rays = 8) + sage: K.contains(random_element(K)) + True + + """ + V = K.lattice().vector_space() + F = V.base_ring() + coefficients = [ F.random_element().abs() for i in range(K.nrays()) ] + vector_gens = map(V, K.rays()) + scaled_gens = [ coefficients[i]*vector_gens[i] + for i in range(len(vector_gens)) ] + + # Make sure we return a vector. Without the coercion, we might + # return ``0`` when ``K`` has no rays. + v = V(sum(scaled_gens)) + return v + + +def positive_operators(K): + r""" + Compute generators of the cone of positive operators on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``P`` in the list should have the property that ``P*x`` + is an element of ``K`` whenever ``x`` is an element of + ``K``. Moreover, any nonnegative linear combination of these + matrices shares the same property. + + EXAMPLES: + + The trivial cone in a trivial space has no positive operators:: + + sage: K = Cone([], ToricLattice(0)) + sage: positive_operators(K) + [] + + Positive operators on the nonnegative orthant are nonnegative matrices:: + + sage: K = Cone([(1,)]) + sage: positive_operators(K) + [[1]] + + sage: K = Cone([(1,0),(0,1)]) + sage: positive_operators(K) + [ + [1 0] [0 1] [0 0] [0 0] + [0 0], [0 0], [1 0], [0 1] + ] + + Every operator is positive on the ambient vector space:: + + sage: K = Cone([(1,),(-1,)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [[1], [-1]] + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: positive_operators(K) + [ + [1 0] [-1 0] [0 1] [ 0 -1] [0 0] [ 0 0] [0 0] [ 0 0] + [0 0], [ 0 0], [0 0], [ 0 0], [1 0], [-1 0], [0 1], [ 0 -1] + ] + + TESTS: + + A positive operator on a cone should send its generators into the cone:: + + sage: K = random_cone(max_ambient_dim = 6) + sage: pi_of_K = positive_operators(K) + sage: all([K.contains(p*x) for p in pi_of_K for x in K.rays()]) + True + + """ + # Sage doesn't think matrices are vectors, so we have to convert + # our matrices to vectors explicitly before we can figure out how + # many are linearly-indepenedent. + # + # The space W has the same base ring as V, but dimension + # dim(V)^2. So it has the same dimension as the space of linear + # transformations on V. In other words, it's just the right size + # to create an isomorphism between it and our matrices. + V = K.lattice().vector_space() + W = VectorSpace(V.base_ring(), V.dimension()**2) + + tensor_products = [ s.tensor_product(x) for x in K for s in K.dual() ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the positive operators, expressed as + # long vectors.. + L = ToricLattice(W.dimension()) + pi_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + pi_cone = pi_dual.dual() + + # And finally convert its rays back to matrix representations. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ M(v.list()) for v in pi_cone.rays() ] + + +def Z_transformations(K): + r""" + Compute generators of the cone of Z-transformations on this cone. + + OUTPUT: + + A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``. + Each matrix ``L`` in the list should have the property that + ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the + discrete complementarity set of ``K``. Moreover, any nonnegative + linear combination of these matrices shares the same property. + + EXAMPLES: + + Z-transformations on the nonnegative orthant are just Z-matrices. + That is, matrices whose off-diagonal elements are nonnegative:: + + sage: K = Cone([(1,0),(0,1)]) + sage: Z_transformations(K) + [ + [ 0 -1] [ 0 0] [-1 0] [1 0] [ 0 0] [0 0] + [ 0 0], [-1 0], [ 0 0], [0 0], [ 0 -1], [0 1] + ] + sage: K = Cone([(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)]) + sage: all([ z[i][j] <= 0 for z in Z_transformations(K) + ....: for i in range(z.nrows()) + ....: for j in range(z.ncols()) + ....: if i != j ]) + True + + The trivial cone in a trivial space has no Z-transformations:: + + sage: K = Cone([], ToricLattice(0)) + sage: Z_transformations(K) + [] + + Z-transformations on a subspace are Lyapunov-like and vice-versa:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) + sage: zs = span([ vector(z.list()) for z in Z_transformations(K) ]) + sage: zs == lls + True + + TESTS: + + The Z-property is possessed by every Z-transformation:: + + sage: set_random_seed() + sage: K = random_cone(max_ambient_dim = 6) + sage: Z_of_K = Z_transformations(K) + sage: dcs = K.discrete_complementarity_set() + sage: all([(z*x).inner_product(s) <= 0 for z in Z_of_K + ....: for (x,s) in dcs]) + True + + The lineality space of Z is LL:: + + sage: set_random_seed() + sage: K = random_cone(min_ambient_dim = 1, max_ambient_dim = 6) + sage: lls = span([ vector(l.list()) for l in K.lyapunov_like_basis() ]) + sage: z_cone = Cone([ z.list() for z in Z_transformations(K) ]) + sage: z_cone.linear_subspace() == lls + True + + """ + # Sage doesn't think matrices are vectors, so we have to convert + # our matrices to vectors explicitly before we can figure out how + # many are linearly-indepenedent. + # + # The space W has the same base ring as V, but dimension + # dim(V)^2. So it has the same dimension as the space of linear + # transformations on V. In other words, it's just the right size + # to create an isomorphism between it and our matrices. + V = K.lattice().vector_space() + W = VectorSpace(V.base_ring(), V.dimension()**2) + + C_of_K = K.discrete_complementarity_set() + tensor_products = [ s.tensor_product(x) for (x,s) in C_of_K ] + + # Turn our matrices into long vectors... + vectors = [ W(m.list()) for m in tensor_products ] + + # Create the *dual* cone of the cross-positive operators, + # expressed as long vectors.. + L = ToricLattice(W.dimension()) + Sigma_dual = Cone(vectors, lattice=L) + + # Now compute the desired cone from its dual... + Sigma_cone = Sigma_dual.dual() + + # And finally convert its rays back to matrix representations. + # But first, make them negative, so we get Z-transformations and + # not cross-positive ones. + M = MatrixSpace(V.base_ring(), V.dimension()) + + return [ -M(v.list()) for v in Sigma_cone.rays() ]